Issue 
Acta Acust.
Volume 8, 2024



Article Number  34  
Number of page(s)  16  
Section  Virtual Acoustics  
DOI  https://doi.org/10.1051/aacus/2024023  
Published online  09 September 2024 
Scientific Article
Diffuse sound field synthesis: Ideal source layers
University of Music and Performing Arts, Institute of Electronic Music and Acoustics, Inffeldgasse 10/3, 8020 Graz, Austria
^{*} Corresponding author: zotter@iem.at
Received:
20
September
2023
Accepted:
6
June
2024
Can uncorrelated surrounding sound sources be used to generate extended diffuse sound fields? By definition, targets are a constant sound pressure level, a vanishing active sound intensity, and uncorrelated sound waves arriving isotropically from all directions. Are there ideal source layouts to synthesize a maximum diffuse sound field within? As methods, we employ numeric simulations and undertake a series of considerations based on uncorrelated source layouts at a finite radius. Statistically expected active sound intensity and sound energy density are insightful and highlight the relation of active sound intensity to potential theory. Correspondingly, both Gauß’ divergence and Newton’s spherical shell theorem apply, and they provide valuable insights. In a circular layout, uncorrelated elementary pointsource fields decaying by 1/√r ideally compose an extended sound field of vanishing active sound intensity; in spherical layouts this is the case with a 1/r decay. None of the layouts synthesizes a perfectly constant sound energy density inside. Theory and simulation offer a broad basis for understanding the synthesis of diffuse sound fields with uncorrelated sources in the free sound field.
Key words: Sound field synthesis / Diffuse sound fields / Gauß’ divergence theorem / Newton’s spherical shell theorem / Potential theory
© The Author(s), Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The concept of a diffuse sound field refers to uncorrelated unitvariance plane waves impinging from all directions. In an ‘ideal’, cylindrically or spherically isotropic diffuse field, the sound pressure level is positionindependent and the timeaveraged sound intensity vector vanishes at all positions [1]. Measurement of sound field diffuseness has an extensive history in acoustics, for instance based on the welldefined frequencydependent correlation between two points in space [2–5]. In addition, the directional distribution of the incident sound [6] or the sound intensity vector can be used to quantify the diffuseness of the sound field [7–12]. In particular, sound intensity normalized by the energy density and speed of sound were defined as energy vector [13, 14] when the diffuse sound field is synthesized by incoherently driven loudspeakers, as it simplifies to the gainsquareweighted average of the loudspeaker directions. The resulting vector is of unit length for fully directional sound fields and zero length for fully diffuse sound fields. More recent approaches employed spherical microphone arrays to measure the diffuseness of the sound field, e.g. COMEDIE, which is based on an eigenvalue decomposition of the spherical harmonics (SH) covariance matrix [15]. A directional and temporal histogram of the energy decay in a simulated room was proposed in auralization [16, 17]. Several recent works moreover showed ways to retrieve or represent anisotropic reverberation properties based on measurements [18–30].
Synthesis of diffuse sound fields with loudspeaker arrays is of interest in laboratory environments, for example to quantify the diffusefield sound transmission of building partitions [31] or acoustic absorption [32, 33], the diffusefield response of microphone arrays [34–36] and dummy heads [37–39], or to measure the performance and perceptual quality of active noise control/cancellation algorithms [40–42], etc.
Spatial sound reproduction targets a faithful reproduction of diffuse sound fields that perceptually elicits a sensation of envelopment for the audience, by suitably employing surrounding loudspeakers [43–47].
Theoretically, sound field synthesis techniques such as wave field synthesis (WFS) [48–55] or nearfield compensated higherorder Ambisonics (NFCHOA) [52, 55–59] should allow to reproduce an ideally diffuse sound field, given a continuous distribution of elementary sources. In discretedirection implementations, however, spatial aliasing severely affects the sound field synthesis capacities of both WFS and HOA in much of the auditory range. Nevertheless, it could be argued that the fruitfulness of these approaches lies in much more tolerant psychoacoustic effects of auditory localization [60], so that playback qualities allow for a much larger sweet area, as analyzed in experiments and perceptual measures [61–65].
In WFS, it was studied for diffusefield synthesis how many uncorrelated planewave sources are needed for a centered listener [66, 67]. In Ambisonics, the required directional resolution or Ambisonic order for a large sweet area was investigated [68], and a directionally nonuniform reverberation algorithm was suggested [69].
The minimum number of loudspeakers and horizontal arrangements for reproducing the spatial impression of diffuse sound fields in channelbased playback was investigated in terms of interaural cross correlation [70] and listening experiments [44] with uncorrelated signals.
Interaural coherence and interaural level differences were moreover considered to evaluate the quality of diffusefield reproduction capabilities [43, 46, 71].
Adding height loudspeaker layers slightly increases perceived diffuseness, when diffuseness is defined as ‘the sound coming from all directions with equal intensity. Therefore, the sound should ideally be impossible to localise and without any gaps [...] in three dimensions’ as shown in [45, 72]. Other researchers showed that two distinct spatial attributes related to the perception of diffuse sound fields can be separated, namely ‘envelopment’ for being surrounded by sound and ‘engulfment’ for being covered by sound [47, 73, 74]. Isotropy as technical measure of directional uniformity was recently discussed [22], and spherical tdesigns were shown to be optimal planewave layouts to synthesize isotropic sound fields [75].
Reverberant spatial audio objects and diffusefield modeling were recently proposed [26, 27] to model the typical/anisotropic characteristics of acoustic rooms. Bandpassbased decorrelation for Diffuse Field Modeling that improves diffuseness in measured 1storder Ambisonic room impulse responses was proposed. By manipulation of the directional weights for front, left, and right, the perception of envelopment was shown to be affected to some degree by direction dependent diffuse fields [26]. Further investigating the perceivability of anisotropy in reverberation, i.e., by recognizing a rotation of the scene, revealed that a time averaged interaural energy decay difference at 850 Hz of slightly more than 1 dB can already be heard [25]. Simplified auralization of nonuniform room reverberation was moreover proposed and tested [76], using a warped, sparse layout of virtual reverberation sources.
In loudspeakerbased reproduction [77], channelbased, scenebased (Ambisonics), and objectbased amplitude panning concepts mostly employ pointsource loudspeakers for playback, yielding a sound pressure decay with distance and parallax with regard to the listening position. Objectbased WFS loudspeaker systems are typically 2.5D and based on dense horizontal pointsource loudspeaker arrays. When rendering planewave sound objects, they yield a sound pressure decay with distance, depending on the listening position, cf. [52, 55]. Surprisingly, all typical loudspeakerbased reproduction still exhibits pronounced shortcomings when used to synthesize extended perceptually diffuse, and therefore consistently nondirectional, sound fields:
Objectbased rendering using amplitude panning, channelbased rendering, or scenebased Ambisonic rendering on moderately dense loudspeaker layouts can be interpreted to exhibit a notoriously small sweet area regarding their capacity of rendering uniformly diffuse listener envelopment. Recent studies revealed that instead of the conventional pointsource loudspeakers, vertical linesource loudspeakers should be able to supply a larger sweet area with a perceptually diffuse sound field [78–81]. With conventional pointsource loudspeakers, the directional impression would often be dominated by the loudspeakers closest to the offcenter listener location, even if the offcenter displacement is limited to, say, a third of the layout radius [80].
A lot of research drive behind WFS was fueled by the idea that its many loudspeakers and wellcontrolled time delays avoid a confined sweet area by shaping consistent and extended wave fronts [60]. Plane waves were thought of as an ideal virtual source type to consistently supply a large listening area. And yet, experiments using 2.5D WFS with uncorrelated virtual plane waves showed that the result was only perceived as nondirectional when the 2.5D reference line correcting the synthesized planewave levels was specific to the listening position [82]. So also 2.5D wavefield synthesis appears limited in rendering a perceptually diffuse sound field to an extended audience. WFS theory suggests [49, 50, 52, 54, 83] that vertical line sources loudspeakers are the optimal sound sources for 2D WFS, of which the generated sound field is constant along the vertical dimension.
A systematic theory is desirable, and might contribute to a profoundly better handling and understanding of diffuse sound field synthesis that may serve many application areas, such as spatial audio and technical acoustics.
1.1 Background and motivation
A noteworthy quote by Jacobsen and Roisin describes diffuse sound fields [84] in the common understanding, based on sources at a large distance:
[…M]ost acousticians would agree on a definition that involves sound coming from all directions. This leads to the concept of a sound field in an unbounded medium generated by distant, uncorrelated sources of random noise evenly distributed over all directions. Since the sources are uncorrelated there would be no interference phenomena in such a sound field, and the field would therefore be completely homogeneous and isotropic. For example, the sound pressure level would be the same at all positions, and temporal correlation functions between linear quantities measured at two points would depend only on the distance between the two points. The timeaveraged sound intensity would be zero at all positions. An approximation to this ‘‘perfectly diffuse sound field’’ might be generated by a number of loudspeakers driven with uncorrelated noise in a large anechoic room […]
These properties summarize the common desires and ideals, and for diffuse sound field synthesis with loudspeakers in the free field, it would be interesting to maximize the size of synthesis that approximates these ideals.
This article is devoted to defining ideal continuously surrounding source layers of uncorrelated sources with uniform strength, whose interior spans a field of finite diameter. In particular, relations to potential theory will be exploited to define such ideal layers and their properties. To motivate the underlying research questions, we first explore the properties of some basic source layouts. As common metrics, cf. [8, 20, 85, 86], we employ potential sound energy density w, active sound intensity I, and diffuseness ψ in the sound field, which are described by the sound pressure p and sound particle velocity vector v,
with the density of air , the speed of sound , and the statistical expectation E{⋅}. In a perfectly diffuse sound field, these quantities are expected to be
Otherwise, diffuseness becomes ψ = 0 for a nondiffuse sound field from a single source.
To observe whether w as a measure of the sound pressure level is constant and the active sound intensity I vanishes, we present a small simulation study motivating the particular research questions of this paper.
Figure 1 shows a simulated freefield map of the sound intensity (vectors), the diffuseness (colors), and the sound energy density (contours) along the horizontal plane, for circular or spherical surfaces S of acoustic sources radiating uncorrelated signals of equal variance from many uniformly distributed directions −u as in [14]. This simplifies the integrals of Section 3.1 to
and ψ is determined by equation (2). The Green’s function decays with for a line source and G_{l} ∝ 1/r_{l} for a point source, cf. equation (13). Distance r_{l} and normalized direction vector u_{l} are measured from each point or line source to the receiver, and normalizes 2ρc^{2}w(0) = 1. Point sources yield the mapping in Figure 1a when arranged in parallel lines and Figure 1c when arranged in a circle, vertical line sources arranged similarly yield the mappings in Figures 1b and 1d, respectively.
Figure 1 Horizontal map of potential sound energy density (contours), normalized active intensity I/(2 c w) (arrows), and diffuseness w with 200+200 point sources (a) or vertical line sources (b) equally spaced along parallel lines of a 6:1 length:distance ratio, 100 point sources (c) or vertical line sources (d) spaced by 3.6° in azimuth for a circle (salmon), and with a sphere of 480 uncorrelated point sources (e) at Chebyshevtype nodes from [87]; all driven by statistically independent signals of uniform variance. 
The parallel and circular arrangements of point sources Figures 1a and 1c appear to be incapable of rendering a uniformly high diffuseness, and their active sound intensity will be dominated by the nearest source. Potential sound energy density or sound pressure level also increases near the sources. By contrast, the parallel and circular arrangements Figures 1b and 1d of vertical line sources and the spherical arrangement Figures 1e of point sources yield a perfectly vanishing active intensity (or diffuseness of ψ = 100%) everywhere inside, while potential sound energy density w still slightly increase towards the sources.
These simulated fields motivate our research questions about diffuse sound field synthesis with a surrounding layer of uncorrelated sources at finite distance:
Can we formulate conditions that cause a vanishing active sound intensity everywhere inside?
Can there be isotropy at every observation point?
Is a constant sound pressure level achievable?
Can twopoint correlations depend only on spacing?
1.2 Outline
Section 2 reviews the common (spherically) isotropic diffuse field model based on uncorrelated plane waves in D dimensions and proves its abovementioned properties. Section 3 introduces uncorrelated source distributions at a finite distance, statistical expectations for active sound intensity and potential sound energy density, and Gauß’ divergence theorem that would measure the total enclosed variance of all sources. The expected intensity reveals a relation to the solution of Green’s functions of potential fields (gravity, electrostatics) [88, 89].
Section 4 deals with question 1, and Gauß’ divergence will be used to prove that the expected active sound intensity can be forced to vanish inside ideal hollow source layers, e.g., a sphere or infinite cylinder in Section 4.3.
Expanding on question 1 and touching question 3, Section 4.7 shows the formal equivalence of the problem to Newton’s spherical shell theorem [90, Prop. 70, Sect. 12] extended to a generalized number of space dimensions. It helps to understand the required balance of oppositedirection active intensity contributions that cancel for any direction, at any observer location.
Concerning question 2 on isotropy, Section 5 uses the integration elements of Newton’s spherical shell theorem to investigate whether isotropic sectoral contributions to active sound intensity are obtained at every observer position inside the hull.
To deal with question 3, Section 6 establishes the scalar potential of active intensity because it is a gradient field in the specific application. It must be constant for ideal layers, and comparison to potential sound energy density shows whether it can be constant.
Section 7 investigates question 4 about whether the correlation inside only depends on distance.
2 Isotropic diffuse field model
This section reviews the typical isotropic, diffuse planewave sound field model and demonstrates its abovementioned properties of constant sound pressure level, vanishing timeaveraged (active) intensity, and pointtopointdistance dependent correlation, everywhere.
A single plane wave from the direction denoted by the unit vector u contributes to the sound pressure p(x) observed at the position x and to the sound particle velocity v(x) observed there; is the wave number, ω = 2πf contains the frequency f, and the imaginary constant is .
Weighted by an amplitude q(u) and superimposed for all surrounding direction vectors in D dimensions, where denotes the Ddimensional unit sphere, the isotropic diffuse field is modeled by these integrals
The abovementioned properties are easily demonstrated when the amplitudes stem from Gaussian random processes q(u) of isotropic variance E{q(u)^{2}} = σ^{2} that are uncorrelated for any pair of different directions
The Dirac delta δ(u_{1 –} u_{2}) is zero unless u_{1} = u_{2}, normalized when integrated over the directions, and possesses the sifting property .
The active intensity vanishes at any position x, using for simplification and equation (7),

The expected sound pressure level denoted as 2ρc^{2}w = E{p^{2}} is constant with regard to the position x, using equation (7) with σ^{2} = 1,
which is the area of the unit sphere , , or for the specific numbers of space dimensions D = 1, 2, 3, or in general , cf. [91, vol.2, p.387], with the gamma function Γ(⋅), [92, Ch.5].
Finally, the position x also cancels from the correlation of the sound pressure p_{1} = p(x), p_{2} = p(x + Δx) at two points because of equation (7), so that correlation only depends on distance Δx = ∥Δx∥ and becomes as derived for arbitrary dimensions D in Appendix A,
or as summarized in Table 1 using the specific functions for D = 1, 2, 3. The higher the number of dimensions D gets that are used for the isotropic distribution of planewave directions, the faster correlation decays over kΔx. Whereas the onedimensional case D = 1 may be uninteresting for most applications, it is typical to regard the D = 2, 3 cases, cf. [43].
Correlation function C(Δx) of the sound pressure at two points spaced by Δx in the ideally isotropic diffuse sound field of D = 1, 2, 3 space dimensions.
3 Distributed, uncorrelated sources
A sound field p(x) that we aim to generate shall be superimposed from an amplitude distribution q(x) that acts as a source term to excite the Helmholtz equation, cf. [93],
with the Laplacian denoting the second order Cartesian derivatives for all coordinates x_{i}.
Typically, the inhomogeneous problem is solved via the Green’s function of the free field, cf. [93],
which is the wave of an excitation at a single source point x_{s} to which r=∥x − x_{s}∥ is the Euclidean distance. This excitation is expressed by the Dirac delta that is zero everywhere else than x = x_{s}, normalized , and sifting . The freefield Green’s function must obey Sommerfeld’s radiation condition [94, §28], see derivations in Appendix B and [94, Ch.V(A.IV,C), E13], using the Hankel function of the second kind [92, 10.2.E5], Table 2. The freefield Green’s function is
with the amplitude . The sound pressure p(x) is obtained by superimposing Green’s functions G weighted by q(x_{s}) for every source point: Expanding the righthand side of equation (11) as implies with equation (12) the solution on the lefthand side of equation (11)
Green’s function G(r) of the Helmholtz equation in D = 1, 2, 3 space dimensions.
The sound particle velocity is , and accordingly
3.1 Expected energy density and intensity
The active intensity is, cf. Section B.4 equation (B9),
The connecting line between any source point x_{s} and any point of observation x defines a unitlength direction vector .
The potential sound energy density 2ρc^{2}w = E{p^{2}} scaled by 22ρc^{2} equals the expected magnitudesquare sound pressure, cf. Section B.5 equation (B12),
These volume integrals equation (16) and equation (17) will only be reduced later to a thin surface layer in equation (24) of Section 4.2 and equation (33) of Section 6, respectively.
4 Ideal layout for zero active intensity
This section shows how ideal source geometries of uniform variance are able to enforce an active intensity that vanishes inside a volume of observation.
4.1 Gauß’ law and sound power
Sound power P_{ac} is typically defined by integrating the divergence of the active intensity over the source volume, e.g. [85], , and Gauß’ law allows to determine it by alternatively integrating the intensity flux aligned with the unitlength surface normal vector n out of the hull,
We inspect the integrand that describes the contribution to active intensity in equation (16). One can relate it to the gradient of the Green’s function of the potential equation, see Appendix C and Table 3 or [94, Ch. II.E],
Green’s function of the potential equation for D = 1, 2, 3 space dimensions and their derivative.
Since is normalized by the area of the unit sphere in D dimensions , and the gradient of uses , we find
which shows that equation (16) yields a gradient field. Divergence of the gradient determines the integrand of equation (18), by which we solve the integral
Upon insertion, it yields the total sound power of the incoherent sources enclosed in V_{s} by Gauß’ law equation (18),
that is wellknown for D = 3: .
4.2 Surrounding thinshell source layer
In this article, we will work with another observed volume V ∉ V_{s} that is free of sources inside, so that the sound power P_{ac} vanishes inside, cf. Figure 2,
Figure 2 Layout of uncorrelated source shell S. 
We moreover reduce the volume source distribution in V_{s} to a thin shell S ⊂ V_{s} of sources distributed tightly around the observed volume V, but not contained within yet, V_{s}, S ⊈ V, so that the interior remains sourcefree. Reduction of the volume integral turns the active sound intensity equation (16) into a surface integral, using equation (20),
with a factor for normalization. The surface element dS is enclosed in an infinitesimally small angular cone dΩ around u, intersecting with the surface normal n at an angle ϕ. The surface element scales by the reciprocal of the projection cosϕ = n^{⊺}u and by a distancerelated area increase r^{D−1} so that , cf. Figure 2.
4.3 Ideal thinshell geometry
The sourcefree interior of the volume V as in equation (23) can only be exploited under geometry assumptions determining the active intensity in equation (23) and forcing it to zero everywhere in V. If, as in Figure 3, active intensity:
has a uniform expected magnitude at every surface position ∥I(x_{s})∥ = I_{n} ∀x_{s} ∈ ∂V and
is nontangential, so aligns with the surface normal at every position I(x_{s}) = I_{n}n(x_{s}) ∀x_{s} ∈ ∂V
Figure 3 Rotationinvariant and shiftinvariant symmetries enforce intensity aligned with surface normal n. 
then it must vanish on ∂V due to the sourcefree interior (P_{ac} = 0) and a nonzero area ()
which causes a zero active intensity I ≡ 0 ∀x ∈ V.
Proof: I can be defined as gradient ρcI = −∇U_{I} of a potential U_{I} due to equations (16) and (20). It is sourcefree in V cf. equation (23) so that △U_{I} = 0 in V, and for the product U_{I}∇U_{I}, Gauß’ divergence theorem/Green’s first identity yields
where assuming I_{n} ≡ 0 on yields , whose proportion to can only hold if also I vanishes uniformly inside V.
The constraints for I(x_{s}) require rotation invariance and shift invariance of the expected amplitude variance σ^{2}(x_{s}). Rotation invariance σ^{2} (Rx_{s}) = σ^{2}(x_{s}) uses a rotation matrix R. It may be limited to a subspace of dimensions if there is shift invariance σ^{2}(x_{s}) = σ^{2}(x_{s} + Δx) with regard to the other E = D – D‣ dimensions σ^{2}(x_{s}) = σ^{2}(x_{s} + Δx), cf. Figure 3.
The discussion of this section assumes incoherent sources for all the D space dimensions that are occupied.
4.4 Ideal source layer for three dimensions
For D = 3, ideal source coordinates lie at a constant radius and describe with D′ = 3, 2, 1 rotationinvariant plus complementing E = 0, 1, 2 shiftinvariant dimensions for σ^{2}:
0: a sphere, ,
1: an infinite cylinder, e.g., ,
2: a pair of opposing infinite planes, e.g., .
This first case relates to Figure 1e that demonstrated the sphere of uncorrelated point sources to yield an ideally vanishing active intensity.
The first and second case explain why a ring of point sources in Figure 1c is not ideal and may produce a nonzero active intensity: an ideal threedimensional layout involving a horizontal ring either needs to be a sphere or an infinite cylinder of uncorrelated point sources. Similarly the parallel lines of point sources in Figure 1a lack vertical extent and are therefore not ideal.
4.5 Ideal source layer for two dimensions
In D = 2space dimensions, ideal source layers with constant radius may use D′ = 2,1 rotationinvariant dimensions when complemented with E = 0, 1 shiftinvariant dimensions for σ^{2}:
0: a circle, ,
1: a pair of opposing infinite lines, e.g., .
While cases for fewer than three dimensions may seem practically irrelvant, they can be quite useful, in fact. The first case demonstrates that the circle of uncorrelated vertical line sources in Figure 1d is an ideal layout: Its linear sound field quantities are independent of the vertical z coordinate, therefore imply vertically coherent line sources. Similarly, the second case of opposing parallel lines confirms the ideal condition in Figure 1b.
4.6 Ideal source layer for one dimension
In D = 1 space dimensions, only the E = 0 case remains with flip invariance for σ^{2}:
0: a pair of opposing points, e.g., .
While this case appears to be the least valuable one, by only using two opposing plane waves that are uncorrelated, it still holds valuable insights and is perfectly 1Disotropic: It yields a vanishing active intensity, a uniform sound pressure level, and its correlation C = cos(kΔx) is ideal by only depending on kΔx, cf. Table 1.
4.7 Newton’s spherical shell theorem
Newton’s original spherical shell theorem [90, Prop. 70, Sect. 12] proves that the gravitational force caused by a thin hollow spherical shell of mass vanishes at every position x enclosed. While essentially equivalent to the symmetry assumptions above, it is geometrically insightful. The contribution dF of a surface element dS at x_{s} to the force at x is proportional to the factor (inverse square law for D = 3 of ) of its distance r = ∥x − x_{s}∥ times , related to the angular element dΩ, as described above. The force contributions dF_{−} and dF_{+} of opposing intersections along the directions ±u of any straight line annihilate
whenever both intersection angles match ϕ_{+} = ϕ_{−}. For any x, this the case for any opposing pair of intersection points and on a hyperspherical manifold, by the isosceles triangle enclosed with the manifold’s origin x_{i} = 0 ∀ i ≤ D′, and more obviously for parallel layers, cf. Figure 4 (middle, bottom). With this match for any ±u, integration over all directions u yields a vanishing net force F = 0. Despite a global sign difference, dI = −dF, the formulation for intensity is equivalent.
Figure 4 In Newton’s spherical shell theorem, intersection angles only match ϕ_{+} = ϕ_{−} under rotation (middle) or shift (bottom) invariance, not in general (top). 
Assuming symmetry ϕ_{+} = ϕ_{−} and the closer side of the hull along u at the distance r_{+} and the longer distance r_{−} to the opposite side, Newton’s theorem shows that a modified source decay 1/r^{2β} may cause imbalance: Contributions and are exactly balanced with the decay 2β = D − 1. Larger exponents cause a dominance of the closest side dI_{+}, smaller ones of the more distant one dI_{−}.
5 Why isotropy may fail offcenter
Isotropy was expressed as the uniformity of the magnitude for all planewave directions received [19], [6, here: Richtungsdiffusität], or their spherical wavenumber spectra [22]. Equivalently, we may assess isotropy via the sectoral active intensity from an infinitesimal angular cone dΩ aligned with a variable direction u, corresponding to an equation side of equation (27), but with arbitrary σ^{2},
so that isotropy can be measured by how constant
evolves across directions u. For σ^{2} =1, isotropy is trivial in the isotropic model Section 2 with D = D′, R_{s} → ∞ or for D = 1, since the direction cosine is unity cosϕ ≡ 1. Otherwise, isotropy is a stricter constraint than Newton’s spherical shell theorem that just requires for every direction u the annihilation with the opposing direction −u by . The strictness of isotropy lies in directionindependence, which is only accomplished (i) at the center of a perfectly spherical layout, D ′= D, σ^{2} =1, r = R_{s} = const, and cosϕ ≡ 1, or (ii) at a particular point of observation within an arbitrary convex layout with a source variance
for every direction u. Assuming to accomplish perfect isotropy by this fixed σ^{2}(u_{s}) that matches the respective direction cosine cosϕ_{0} at every source to a central observer at x = 0, an observer shifted off center will yield a different direction cosine cosϕ at each source, in general, contradicting the assumption. No fixed choice of σ^{2} is therefore able to enforce isotropy everywhere inside V, neither in an arbitrary convex nor spherical case (ii).
For uniform variance, σ^{2} = 1, Figure 5 shows the sectoral intensity for the circular/spherical and parallel cases with D = 2,3 with regard to a radial shift x = {0, 0.71, 0.87} of the observer. For the parallels, the peak at ϕ = ϑ = 90° is infinite and positionindependent as the direction cosine cosϕ in the denominator always vanishes there; its shiftinvariant infinite dimensions dominate sectoral intensity. In the circular/spherical case, sectoral intensity peaks at ϑ = 90°. It is found by and the law of sines of a triangle with sides and opposite angles: shift x and surface angle ϕ, radius R_{s} and polar angle ϑ.
Figure 5 Sectoral intensity to assess isotropy for the 2D circle/3D sphere case (solid) and the 2D parallel line and 3D parallel plane cases (dashed) across observation angle enclosed with the axis of a radial shift of 0% (green), 71% (orange), 87% (blue) of the observer. 
6 Why level may not be constant
Because it accomplishes the synthesis of vanishing active sound intensity, we assume a thin source shell in the space and fix the variance to σ^{2} = 1 at some radius, e.g. R_{s} = 1 of the unit sphere, denoted as . While both equation (25) and Newton’s spherical shell theorem equation (27) were proof that (hyper) spherical shells are an ideal geometry for vanishing active intensity,
this section shows why this may not yield an ideally constant potential soundenergy density w, equation (2), or sound pressure level E{p^{2}}, as displayed in Figures 1b–1e in terms of a slight level increase towards the sources.
The surface element is of an isotropic angular segment dΩ_{0} seen from the origin, the area of the unit sphere is for one less space dimensions (as above: , cf. [91, vol.2, p.387]), the normalizer is , and the subscript D is used to track the number of dimensions.
To utilize the formalism from above for the potential sound energy density w, we employ highfrequency or farfield approximation . It permits a similar discussion as for active intensity, by relating the Green’s functions of the Helmholtz and potential equations everywhere for D = 1, 3 otherwise for large kr,
cf. Section B.5. For D ≠ 1, 3 the approximation is asymptotic for high frequencies or large distances satisfying . For D = 2, where , a distance to the closest source as small as is approximated well for frequencies above . With equation (32), reduction of the volume integral equation (17) to a hyperspherical shell integral yields
It may only be constant for D = 1, as a trivial case with , cf. Table 3. We use the formal similarity to equation (31) to show why.
Since the gradient field equation (31) of active intensity is not yet a scalar counterpart to equation (33), we make use of the potential U_{I} to which it relates with −∇U_{I} = ρcI and which must be constant inside V because I = 0,
The function involved in the potential U_{I} of equation (34) differs from involved in 2ρc^{2}w of equation (33). We know of to provide a constant U_{I} . Analogy to the discussion in Section 4.7 shows that potential sound energy density 2ρc^{2}w won’t be constant: Its exponent for 1/r is too high, which yields an imbalance towards the closest, dominating sources of the surrounding layer (with the exception of D = 1).
Numerical results with settings as in Section 7 show this increase in Figure 6 for the ideal layers with the 1D case as an exception, and the amount of the level increase reduces by the number of shiftinvariant dimensions.
Figure 6 Outwards increase of the sound pressure level for the ideal layers (1D: opposing sources, 2D: parallel lines, circle, 3D: opposing planes, cylinder, sphere). 
7 Why correlation may be anisotropic
Correlation of the sound pressure at two points and is affected by interference, unlike the quadratic superposition of I and w,
where . Using and for high frequency and small shift yields , so that
This equation converges to ideal isotropy of Section 2 equation (10) only if D = D′ and R_{s} → ∞ so that cosϕ → 1.
We evaluate equation (35) for ideal layouts, σ^{2} = 1, R_{s} = 1, at a center x = 0 and an offcenter radial position x = [0.87, 0,0]⊺, for radial , tangential , axial shifts separated by Δx = 0.1, with Δs ≈ 0.01 to discretize the integrals. Parallel lines in 2D, Figure 7a, yield a tangential correlation between cos(kΔx) and J_{0}(kΔx), nearly positionindependently. Radial correlation is shallower, not as regular, and positiondependent; numerical integration summed up 2000 × 2 sources on 2 lines within y_{s} ∈ [−10; 10] at x_{s} = ±1. For the circle in 2D, Figure 7b, correlation approximates J_{0}(kΔx) best at x = 0, and less so at x = 0.87, where radial and tangential correlations differ; numerical integration summed up 630 sources along a unit circle . Parallel planes in 3D, Figure 7c, yield a tangential correlation between J_{0}(kΔx) and that is nearly positionindependent, whereas radial correlation is positiondependent and not as regular; numerical integration summed up 2000 × 2000 × 2 sources on the intervals y_{s}, z_{s} ∈ [−10; 10], at x_{s} = ±1. For the cylinder in 3D, Figure 7d, axial correlation approximates J_{0}(kΔx) nearly positionindependently, while radial and tangential correlations are positiondependent and differ; numerical integration summed up 2000 × 630 sources on the axial interval z_{s} ∈ [−10; 10], along unit circles . For the sphere in 3D, Figure 7e, tangential correlation is nearly positionindependent and approximates , and radial correlation is positiondependent; numerical integration summed up 130000 sources arranged in a uniform Chebyshevtype spherical layout [87, 130000 (random)]. Apparently, using a maximum number of rotationally symmetric dimensions D′ = D best approximates isotropic correlation.
Figure 7 Numerically integrated correlation C(f) for the ideal 2D layers of (a) parallel lines and (b) a circle, the ideal 3D layers of (c) parallel planes, (d) a cylinder, (e) a sphere, in the center x = 0 and offcenter by x = 0.87 with R = 1, r^{2} = 1, and a ∆x = 0.1 of radial (“r”, dotted), axial (“a”, dashdot), or tangential (“t”, dashed) orientation; light to dark solid lines display the ideal isotropic correlations for the dimensions D = 1, 2, 3, cf. Table 1. 
8 Conclusion
In this article we discussed theoretical requirements and limitations of diffuse sound fields synthesized with uncorrelated sources in the free field. The findings are of practical relevance when attempting to produce extended diffuse fields with loudspeakers, either in the anechoic chamber or in typical spatial audio rendering scenarios with loudspeakers or virtual sources. Results demonstrated are accurate in terms of active sound intensity and isotropy for any number of dimensions, and for sound pressure level (potential sound energy density) they are accurate either for D = 1, 3 dimensions or under farfield conditions .
We could prove there being theoretically ideal source layouts that can synthesize vanishing active sound intensity within the entire field enclosed. This was accomplished by statistical expectation and by assuming a continuous layer of uncorrelated surrounding sources of uniform strength. Hereby, the active intensity of superimposed uncorrelated Green’s functions of the Helmholtz equation assumed the form of superimposing the gradient fields of Green’s functions of the potential equation. Potential theory permitted interpretations accessible through Gauß’ divergence theorem and Newton’s spherical shell theorem. Here is a list of ideal layouts:
A uniform spherical layer of uncorrelated point sources is able to produce vanishing sound intensity in 3D, i.e. 100% diffuseness, everywhere inside.
A circular layer of uncorrelated sources produces 100% diffuseness everywhere inside in 2D; an infinite cylindrical source layer is needed in 3D.
Opposing uncorrelated sources yield 100% diffuseness between them in 1D; infinite parallel layers of sources are needed in 2D (lines) and 3D (planes).
We could prove for ideal source layers that sound pressure level is not ideally constant in the entire field enclosed and slightly increases towards the source layer, unless planewave sources are used.
We could prove that the ideal, uncorrelated 2D and 3D source layers cannot provide ideal isotropy at every observer position inside. Direction cosines cosϕ to the surface normal cause anisotropy offcenter for circular 2D or spherical 3D layouts, or cause a dominance of the shiftinvariant dimensions everywhere.
We could moreover prove for the field enclosed by ideal source layers that correlation measured between a pair of points depends not only on distance, but also on the orientation and position, in particular when there is high anisotropy.
Future work should try to solve the integrals involved analytically, find optimal source variances for less ideal geometries (e.g. elliptic, cuboid, and hemispherical layouts) and investigate the diffuseness limits of discrete instead of continuous source layers. Moreover, properties of 2.5D WFS as a method to synthesize virtual source arrangements should be investigated. Modified sources should be used to explore options for the nonideal sound pressure level, isotropy, and correlation metrics.
Acknowledgments
We gratefully received funding from the Austrian Science Fund (FWF): P 35254N (Envelopment in Immersive Sound Reinforcement, EnImSo). We thank the editor and anonymous reviewers for their appreciation of the topic and guidance through substantial revisions.
Conflict of interest
The authors declare no conflict of interest
Data availability statement
The research data associated with this article are available in the iem git repository, under the reference [95].
Appendix A
Correlation in the diffuse field
The correlation in the Ddimensional, ideally isotropic diffuse sound field uses the integral presented in equation (10) of Section 2 yields
which was simplified by the rotationally symmetric integration element , where z = cosϑ is the direction cosine with regard to the axis of the shift Δx, and the area of the unit sphere in D dimensions is . Using the integral from [92, 10.9.E4], the equation becomes in general, for D dimensions,
For the particular cases of D = 1,3 dimensions, integration is rather straightforward, D = 2 uses the typical integral definition [92, 10.9.E1] of the Bessel function,
Appendix B
Helmholtz Green’s function
Knowing that the freefield Green’s function only depends on , we may define it via the Laplacian in terms of the radius r, only . The corresponding homogeneous equation simplifies by multiplication with r^{2} and substitution x = kr to
According to Sommerfeld [94, Ch.V(A.IV,C)], transform via the product G = ax^{−ν}v(x) of a suitable power ν of x, a scalar a, and a solution v(x) is useful and leaves after inserting and dividing by ax^{−ν}
B.1 Farfield or D = 1,3 solution
The specific choice 2ν = D − 1 turns equation (41) into the differential equation of an undamped oscillator, v = e^{±ix}), whenever ,
which yields . This is either precise for ν = 0, 1, so D = 1, 3, or it is a farfield approximation for D ≠ 1, 3, otherwise.
B.2 General solution for any
The standard substitution 2ν = D − 2 for a general solution turns equation (B2) into the Bessel differential equation
As a possible solution for v, the Hankel function of the second kind [92, 10.2.E6] yields with G = ax^{−ν}v,
This is the physical solution that fulfills Sommerfeld’s radiation condition [94, §28].
B.3 Scaling for any D
While in equation (B4) with x = kr captures how the Green’s function evolves over any radius r > 0 as homogeneous solution, a suitable scaling a(k) must be found in equation (B4) so that G matches its inhomogeneity −δ(x) in equation (12). This is done by integrating equation (12) over a spherical volume
whose radius r we are allowed to minimize by without changing , so that we can utilize as a simplification the smallradius approximation of the Hankel function , cf. [92, 10.7.E7] as in [94, Ch.V(A.IV,C)]. For G and its derivative , it yields:
First, substitution and 2ν = D − 2 is used to show that vanishes in equation (B5)
Then what remains is integrated using the divergence theorem ,
With the smallradius form from above and , this yields a:
Together with x^{−v}H_{ν}(x), it is used in Section 3 equation (13)
B.4 Velocity and active intensity for any D
The radial velocity of equation (B7) is, using [92, 10.6.E1],
Multiplied with , it yields the active radial intensity of the single source
which was simplified by the Wronskian [92, 10.5.E5] and yields the expression in equation (16) of Section 3.1.
B.5 Farfield potential soundenergy density for any
The potential sound energy density 2ρc^{2}w = p^{2} becomes with equations (B7) and (B8) for the single source,
and with the farfield approximation [92, 10.2.E6]
the farfield Green’s function becomes
and we obtain for the potential sound energy density
These equations are applied in Section 3 equation (17) and Section 6 equation (32), and their common amplitude term is
Appendix C
Potential Green’s function
The Green’s function of the potential equation fulfills
and for , we have a homogeneous equation
that is fulfilled by the power
and using Gauß law, as before, we can find the constant a of the inhomogeneous problem for any r > 0
as used in Section 4 equation (19) and Table 3.
References
 F. Jacobsen: A note on instantaneous and time averaged active and reactive sound intensity. JSV 147, 3 (1991) 489–496. [Online]. Available: https://doi.org/10.1016/0022460X(91)904967. [CrossRef] [Google Scholar]
 R.K. Cook, R. Waterhouse, R. Berendt, S. Edelman, M. Thompson Jr: Measurement of correlation coefficients in reverberant sound fields. JASA 27, 6 (1955) 1072–1077. [Online]. Available: https://doi.org/10.1121/1.1908122. [CrossRef] [Google Scholar]
 C. Balachandran: Random sound field in reverberation chambers. JASA 31, 10 (1959) 1319–1321. [Online]. Available: https://doi.org/10.1121/1.1907626. [CrossRef] [Google Scholar]
 H. Kuttruff: Raumakustische Korrelationsmessungen mit einfachen Mitteln. Acustica 13 (1963) 120–122. [Google Scholar]
 P. Dämmig: Zur Messung der Diffusität von Schallfeldern durch Korrelation. Acta Acustica United with Acustica 7, 6 (1957) 387–398. [Online]. Available: https://www.ingentaconnect.com/content/dav/aaua/1957/00000007/00000006/art00008#. [Google Scholar]
 R. Thiele: Richtungsverteilung und Zeitfolge der Schallrückwürfe in Räumen. Acta Acustica United with Acustica 3, 4 (1953) 291–302. [Online]. Available: https://www.ingentaconnect.com/content/dav/aaua/1953/00000003/a00204s2/art00007. [Google Scholar]
 I. Veit, H. Sander: Production of spatially limited “diffuse” sound field in an anechoic room. JAES 35, 3 (1987) 138–143. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=5219. [Google Scholar]
 V. Pulkki: Spatial sound reproduction with directional audio coding. JAES 55, 6 (2007) 503–516. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=14170. [Google Scholar]
 J. Ahonen, V. Pulkki: Diffuseness estimation using temporal variation of intensity vectors. In: IEEE WASPAA, IEEE, 2009, pp. 285–288. [Online]. Available: https://doi.org/10.1109/ASPAA.2009.5346496. [Google Scholar]
 G. Del Galdo, M. Taseska, O. Thiergart, J. Ahonen, V. Pulkki: The diffuse sound field in energetic analysis. JASA 131, 3 (2012) 2141–2151. [Online]. Available: https://doi.org/10.1121/1.3682064. [CrossRef] [PubMed] [Google Scholar]
 F. Jacobsen: Active and reactive sound intensity in a reverberant sound field. JSV 143, 2 (1990) 231–240. [Online]. Available: https://doi.org/10.1016/0022460X(90)90952V. [CrossRef] [Google Scholar]
 F. Jacobsen: Active and reactive, coherent and incoherent sound fields. JSV 130, 3 (1989) 493–507. [Online]. Available: https://doi.org/10.1016/0022460X(89)900722. [CrossRef] [Google Scholar]
 M.A. Gerzon: General metatheory of auditory localisation. In: 92nd AES Conv., Vienna, March 1992. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=6827. [Google Scholar]
 J. Merimaa: Energetic sound field analysis of stereo and multichannel loudspeaker reproduction. In: 123rd AES Conv., New York, October 2007. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=14315. [Google Scholar]
 N. Epain, C.T. Jin: Spherical harmonic signal covariance and sound field diffuseness. IEEE/ACM TASLP 24, 10 (2016) 1796–1807. [Online]. Available: https://doi.org/10.1109/TASLP.2016.2585862. [Google Scholar]
 D. Schröder: Physically based realtime auralization of interactive virtual environments. Ph.D. dissertation, RWTH Aachen, 2011. [Online]. Available: https://publications.rwthaachen.de/record/50580/files/3875.pdf. [Google Scholar]
 M. Vorländer: Auralization. In: Fundamentals of Acoustics, Modelling, Simulation, Algorithms and Acoustic Virtual Reality. 2nd ed., Springer Nature, Switzerland, 2020. [Online]. Available: https://doi.org/10.1007/9783030512026. [Google Scholar]
 H. Okubo, M. Otani, R. Ikezawa, S. Komiyama, K. Nakabayashi: A system for measuring the directional room acoustical parameters. Applied Acoustics 62, 2 (2001) 203–215. [Online]. Available: https://doi.org/10.1016/S0003682X(00)000566. [CrossRef] [Google Scholar]
 B.N. Gover, J.G. Ryan, M.R. Stinson: Measurements of directional properties of reverberant sound fields in rooms using a spherical microphone array. JASA 116, 4 (2004) 2138–2148. [Online]. Available: https://doi.org/10.1121/1.1787525. [CrossRef] [Google Scholar]
 K. Merimaa, V. Pulkki: Spatial impulse response rendering I: analysis and synthesis. JAES 53, 12 (2005) 1115–1127. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=13401. [Google Scholar]
 S. Tervo, J. Pätynen, A. Kuusinen, T. Lokki: Spatial decomposition method for room impulse responses. JAES 61, 1/2 (2013) 17–28. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=16664. [Google Scholar]
 M. Nolan, E. FernandezGrande, J. Brunskog, C.H. Jeong: A wavenumber approach to quantifying the isotropy of the sound field in reverberant spacesa. JASA 143, 4 (2018) 2514–2526. [Online]. Available: https://doi.org/10.1121/1.5032194. [CrossRef] [PubMed] [Google Scholar]
 M. Berzborn, M. Vorländer: Directional sound field decay analysis in performance spaces. Building Acoustics 28, 3 (2021) 249–263. [Online]. Available: https://doi.org/10.1177/1351010X20984622. [CrossRef] [Google Scholar]
 P. Massé, T. Carpentier, O. Warusfel, M. Noisternig: Denoising directional room impulse responses with spatially anisotropic late reverberation tails. Applied Sciences 10, 3 (2021) 1033. [Online]. Available: https://doi.org/10.3390/app10031033. [Google Scholar]
 B. Alary, P. Massé, S.J.S. anad Markus Noisternig, V. Välimäki: Perceptual analysis of directional late reverberation. JASA 149, 5 (2021) 3189–3199. [Online]. Available: https://doi.org/10.1121/10.0004770. [CrossRef] [PubMed] [Google Scholar]
 D. Romblom: Diffuse field modeling– the physical and perceptual properties of spatialized reverberation. Ph.D. dissertation, McGill University, Montréal, 2016. [Online]. Available: https://escholarship.mcgill.ca/downloads/rf55zb38p. [Google Scholar]
 P. Coleman, A. Franck, P.J.B. Jackson, R.J. Hughes, L. Remaggi, F. Melchior: Objectbased reverberation for spatial audio. JAES 65, 1/2 (2017) 66–77. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=18544. [Google Scholar]
 G. Götz, S.J. Schlecht, V. Pulkki: Commonslope modeling of late reverberation in coupled rooms. In: ICA, Gyeongju, October 2022. [Online]. Available: https://ica2022korea.org/data/Proceedings_A12.pdf. [Google Scholar]
 C. Hold, T. McKenzie, G. Götz,S.J. Schlecht, V. Pulkki: Resynthesis of spatial room impulse response tails with anisotropic multislope decays. JAES 70, 6 (2022) 526–538. [Online]. Available: https://doi.org/10.17743/jaes.2022.0017. [Google Scholar]
 T. Deppisch, S.V.A. Garí, P. Calamia, J. Ahrens: Direct and residual subspace decomposition of spatial room impulse responses. IEEE/ACM TASLP 31 (2023) 927–942. [Online]. Available: https://doi.org/10.1109/TASLP.2023.3240657. [Google Scholar]
 C.V. Hoorickx, E.P. Reynders: Numerical realization of diffuse sound pressure fields using prolate spheroidal wave functions. JASA 151, 3 (2022) 1710–1721. [Online]. Available: https://doi.org/10.1121/10.0009764. [CrossRef] [PubMed] [Google Scholar]
 O. Robin, A. Berry, O. Doutres, N. Atalla: Measurement of the absorption coefficient of sound absorbing materials under a synthesized diffuse acoustic field. JASAEL 136 (2014) EL13–EL19. [Online]. Available: https://doi.org/10.1121/1.4881321. [CrossRef] [PubMed] [Google Scholar]
 S. Dupont, M. Sanalatii, M. Melon, O. Robin, A. Berry, J.C. Le Roux: Measurement of the diffuse field sound absorption using a sound field synthesis method. Acta Acustica 7 (2023) 26. [Online]. Available: https://doi.org/10.1051/aacus/2023021. [CrossRef] [EDP Sciences] [Google Scholar]
 E. Habets, S. Gannot: Generating sensor signals in isotropic noise fields. JASA 122, 6 (2007) 3464–3470. [Online]. Available: https://doi.org/10.1121/1.2799929. [CrossRef] [PubMed] [Google Scholar]
 M. Kustner: Spatial correlation and coherence in reverberant acoustic fields: extension to microphones with arbitrary firstorder directivity. JASA 123, 1 (2008) 152–164. [Online]. Available: https://doi.org/10.1121/1.2812592. [Google Scholar]
 N. Akbar, G. Dickins, M.R.P. Thomas, P. Samarasinghe, T. Abhayapala: A novel method for obtaining diffuse field measurements for microphone calibration. In: IEEE ICASSP, Barcelona, 2020. [Online]. Available: https://doi.org/10.1109/ICASSP40776.2020.9054728. [Google Scholar]
 G. Theile: Comparison of two dummy head systems with due regard to different fields of application. In: DAGA, Darmstadt, 84a0223.pdf, 1984. [Online]. Available: https://pub.degaakustik.de/DAGA.19821990.zip. [Google Scholar]
 T. McKenzie, D.T. Murphy, G. Kearney: Diffusefield equalisation of binaural ambisonic rendering. Applied Sciences 8, 10 (2018) 1956. [Online]. Available: https://doi.org/10.3390/app8101956. [CrossRef] [Google Scholar]
 C. Armstrong, L. Thresh, D. Murphy, G. Kearney: A perceptual evaluation of individual and nonindividual HRTFs: a case study of the SADIE II database. Applied Sciences 8, 11 (2018) 2029. [Online]. Available: https://doi.org/10.3390/app8112029. [CrossRef] [Google Scholar]
 D.J. Moreau, J. Ghan, B. Cazzolato, A. Zander: Active noise control in a pure tone diffuse sound field using virtual sensing. JASA 125, 6 (2009) 3742–3755. [Online]. Available: https://doi.org/10.1121/1.3123404. [CrossRef] [PubMed] [Google Scholar]
 F. Holzmüller, A. Sontacchi: Frequency limitation for optimized perception of local active noise control. In: DAGA, Hamburg, March 2023. [Online]. Available: https://pub.degaakustik.de/DAGA.2023/data/articles/000531.pdf. [Google Scholar]
 S.J. Elliott, P. Joseph, A. Bullmore, P.A. Nelson: Active cancellation at a point in a pure tone diffuse sound field. Journal of Sound and Vibration 120, 1 (1988) 183–189. [Online]. Available: https://doi.org/10.1006/jsvi.1996.0742. [CrossRef] [Google Scholar]
 A. Walther, C. Faller: Assessing diffuse sound field reproduction capabilities of multichannel playback systems. In: 130th AES Conv., London, May 2011. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=15895. [Google Scholar]
 K. Hiyama, S. Komiyama, K. Hamasaki: The minimum number of loudspeakers and its arrangement for reproducing the spatial impression of diffuse sound field. In: 113th AES Conv., Los Angeles, October 2002. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=11272. [Google Scholar]
 M.P. Cousins, F.M. Fazi, S. Bleeck, F. Melchior: Subjective diffuseness in layerbased loudspeaker systems with height. In: 139th AES Conv., New York, October 2015. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=17983. [Google Scholar]
 M. Cousins: The diffuse sound object. Ph.D. dissertation, University of Southampton, 2018. [Online]. Available: https://eprints.soton.ac.uk/442615/1/Thesis_Final_Submitted_19_06_2019.pdf. [Google Scholar]
 S. Riedel, M. Frank, F. Zotter: The effect of temporal and directional density on listener envelopment. JAES 71, 7/8 (2023) 455–467. [Online]. Available: https://doi.org/10.17743/jaes.2022.0088. [Google Scholar]
 A. Berkhout, D. de Vries, P. Vogel: Acoustic control by wave field synthesis. JASA 93, 5 (1993) 2764–2778. [Online]. Available: https://doi.org/10.1121/1.405852. [CrossRef] [Google Scholar]
 E.W. Start: Direct sound enhancement using wave field synthesis. Ph.D. dissertation, TU Delft, 1997. [Online]. Available: https://repository.tudelft.nl/islandora/object/uuid:c80d5b5867d34d849e73390cd30bde0d/datastream/OBJ/download. [Google Scholar]
 E. Verheijen: Sound reproduction by wave field synthesis. Ph.D. dissertation, TU Delft, 1998. [Online]. Available: https://www.dbvision.nl/bestanden/overons/publicaties/ouder/Thesis_Edwin_Verheijen.pdf. [Google Scholar]
 T. Caulkins: Caractérisation et contrôle du rayonnement d’un système de wave field synthesis pour la situation de concert. Ph.D. dissertation, Université de Paris 6, 2007. [Online]. Available: http://architexte.ircam.fr/textes/Caulkins07a/index.pdf. [Google Scholar]
 J. Ahrens: Analytic methods of sound field synthesis, Springer Berlin Heidelberg, 2012. [Online]. Available: https://doi.org/10.1007/9783642257438. [CrossRef] [Google Scholar]
 G. Firtha, P. Fiala, F. Schultz, S. Spors: Improved referencing schemes for 2.5D wave field synthesis driving functions. IEEE/ACM TASLP 25, 5 (2017) 1117–1127. [Online]. Available: https://doi.org/10.1109/TASLP.2017.2689245. [Google Scholar]
 F. Winter: Local sound field synthesis. Ph.D. dissertation, University of Rostock, 2019. [Online]. Available: https://doi.org/10.18453/rosdok_id00002568. [Google Scholar]
 P. Grandjean, A. Berry, P.A. Gauthier: Sound field reproduction by combination of circular and spherical higherorder ambisonics: Part I – a new 2.5D driving function for circular arrays. JAES 69, 3 (2021) 152–165. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=21024. [Google Scholar]
 R. Nicol: Restitution sonore spatialisée sur une zone étendue: Application à la téléprésence. Ph.D. dissertation, Université du Maine, 1999. [Online]. Available: https://theses.hal.science/tel01067541/document. [Google Scholar]
 J. Daniel: Représentation de champs acoustiques, application à la transmission et à la reproduction de scénes sonores complexes dans un contexte multimédia. Ph.D. dissertation, Université de Paris 6, 2001. [Online]. Available: http://gyronymo.free.fr/audio3D/downloads/Theseoriginalversion.zip. [Google Scholar]
 D. Ward, T. Abhayapala: Reproduction of a planewave sound field using an array of loudspeakers. IEEE TASAP 9, 6 (2001) 697–707. [Online]. Available: https://doi.org/10.1109/89.943347. [Google Scholar]
 M.A. Poletti: Threedimensional surround sound systems based on spherical harmonics. JAES 53, 11 (2005) 1004–1025. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=13396. [Google Scholar]
 S. Spors, H. Wierstorf, A. Raake, F. Melchior, M. Frank, F. Zotter: Spatial sound with loudspeakers and its perception: A review of the current state. Proceedings of the IEEE 101, 9 (2013) 1920–1938. [Online]. Available: https://doi.org/10.1109/JPROC.2013.2264784. [CrossRef] [Google Scholar]
 M. Frank: Phantom sources using multiple loudspeakers in the horizontal plane. Ph.D. dissertation, University of Music and Performing Arts Graz, 2013. [Online]. Available: https://phaidra.kug.ac.at/o:7008. [Google Scholar]
 H. Wierstorf: Perceptual assessment of sound field synthesis. Ph.D. dissertation, TU Berlin, 2014. [Online]. Available: https://doi.org/10.14279/depositonce4310. [Google Scholar]
 P. Stitt, S. Bertet, M. van Walstijn: Offcentre localisation performance of ambisonics and hoa for large and small loudspeaker array radii. Acta Acustica United with Acustica 100, 5 (2014) 937–944. [Online]. Available: https://doi.org/10.3813/AAA.918773. [CrossRef] [Google Scholar]
 M. Kuntz, B.U. Seeber: Sound field synthesis: Simulation and evaluation of auralized interaural cues over an extended area. In: Euronoise, Madeira, October 2021. [Online]. Available: https://mediatum.ub.tum.de/doc/1634172/wd5xy0emuqxi7wicws365j5oe.Kun_See_EuroNoise21.pdf. [Google Scholar]
 M. Kuntz, B.U. Seeber: Investigating the smoothness of moving sources reproduced with panning methods. In: DAGA, March, Stuttgart 2022. [Online]. Available: https://pub.degaakustik.de/DAGA_2022/data/articles/000363.pdf. [Google Scholar]
 J.J. Sonke: Variable acoustics by wave field synthesis. Ph.D. dissertation, TU Delft, 2000. [Online]. Available: http://resolver.tudelft.nl/uuid:2039d23c4da340219fb12c21b4cf7275. [Google Scholar]
 J. Ahrens: Perceptual evaluation of the diffuseness of synthetic late reverberation created by wave field synthesis at different listening positions. In: DAGA, Nürnberg, March 2015. [Online]. Available: http://pub.degaakustik.de/DAGA_2015/data/articles/000169.pdf. [Google Scholar]
 M. Frank, F. Zotter: Exploring the perceptual sweet area in ambisonics. In: 142nd AES Conv., Berlin, 2017. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=18604. [Google Scholar]
 B. Alary, A. Politis, S.J. Schlecht, V. Välimäki: Directional feedback delay network. JAES 67, 10 (2019) 752–762. [Online]. Available: https://doi.org/10.17743/jaes.2019.0026. [Google Scholar]
 P. Damaske, Y. Ando: Interaural crosscorrelation for multichannel loudspeaker reproduction. Acta Acustica united with Acustica 27, 4 (1972) 232–238. [Online]. Available: https://www.ingentaconnect.com/content/dav/aaua/1972/00000027/00000004/art00011 [Google Scholar]
 A. Walther, C. Faller: Interaural correlation discrimination from diffuse field reference correlations. JASA 133, 3 (2013) 1496–1502. [Online]. Available: https://doi.org/10.1121/1.4790473. [CrossRef] [PubMed] [Google Scholar]
 M.P. Cousins, S. Bleeck, F. Melchior, F.M. Fazi: Relation between acoustic measurements and the perceived diffuseness of a synthesised sound field. In: Proc ICA, Buenos Aires, September 2016. [Online]. Available: https://eprints.soton.ac.uk/398728/1/Michael_Cousins_ICA_2016_Final.pdf. [Google Scholar]
 S. Riedel, M. Frank, F. Zotter, R. Sazdov: A study on loudspeaker spl decays for envelopment and engulfment across an extended audience. In: AES ASR Conf., Le Mans, Jan 2024. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=22368. [Google Scholar]
 R. Sazdov, G. Paine, K. Stevens: Perceptual investigation into envelopement, spatial clarity, and engulfment in reproduced multichannel audio. In: 31st Int. AES Conf., 2007. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=13961. [Google Scholar]
 T. Tanaka, M. Otani: An isotropic sound field model composed of a finite number of plane waves. Acoustical Science and Technology 44, 4 (2023) 317–327. [Online]. Available: https://doi.org/10.1250/ast.44.317. [CrossRef] [Google Scholar]
 C. Kirsch, T. Wendt, S. van de Par, H. Hu, S.D. Ewert: Computationallyefficient simulation of late reverberation for inhomogeneous boundary conditions and coupled rooms. JAES 17, 4 (2023) 186–201. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=22040. [Google Scholar]
 Study Group 6: Advanced sound system for programme production. Recommendation, Broadcasting Service (sound), no. ITUR BS.20513, 5 2022. [Online]. Available: https://www.itu.int/rec/RRECBS.2051/en. [Google Scholar]
 P. Heidegger, B. Brands, L. Langgartner, M. Frank: Sweet area using ambisonics with simulated line arrays. In: DAGA, Vienna, August 2020. [Online]. Available: https://pub.degaakustik.de/DAGA_2021/data/articles/000374.pdf. [Google Scholar]
 M. Blochberger, F. Zotter, M. Frank: Sweet area size for the envelopment of a recursive and a nonrecursive diffuseness rendering approach. In: ICSA, Ilmenau, 2019, pp. 151–157. [Online]. Available: https://doi.org/10.22032/dbt.39969. [Google Scholar]
 S. Riedel, F. Zotter: Surrounding line sources optimally reproduce diffuse envelopment at offcenter listening positions. JASAEL 2, 9 (2022) 094404. [Online]. Available: https://doi.org/10.1121/10.0014168. [Google Scholar]
 S. Riedel, L. Goelles, M. Frank, F. Zotter: Modeling the listening area of envelopment. In: DAGA, Hamburg, March 2023. [Online]. Available: https://pub.degaakustik.de/DAGA_2023/data/articles/000289.pdf. [Google Scholar]
 F. Melchior, C. Sladeczek, D. de Vries, B. Frohlich: Userdependent optimization of wave field synthesis reproduction for directive sound fields. In: 124th AES Conv., Amsterdam, May 2008. [Online]. Available: http://www.aes.org/elib/browse.cfm?elib=14506. [Google Scholar]
 G. Firtha: A generalized wave field synthesis framework–with application for moving virtual sources. Ph.D. dissertation, Budapest University of Technology and Economics, 2019. [Online]. Available: http://last.hit.bme.hu/download/firtha/PhD_thesis/firtha_phd_thesis.pdf. [Google Scholar]
 F. Jacobsen, T. Roisin: The coherence of reverberant sound fields. JASA 108, 1 (2000) 204–210. [Online]. Available: https://doi.org/10.1121/1.429457. [CrossRef] [PubMed] [Google Scholar]
 H. Kuttruff: Room Acoustics. 6th edn., CRC Press, Boca Raton, 2016. https://doi.org/10.1201/9781315372150 [CrossRef] [Google Scholar]
 F. Fahy: Foundations of Engineering Acoustics, Elsevier Academic Press, San Diego, 2000. https://doi.org/10.1016/B9780122476655.X50000. [Google Scholar]
 M. Gräf: Quadrature rules on manifolds. accessed 2023/04/23. [Online]. Available: https://wwwuser.tuchemnitz.de/~potts/workgroup/graef/quadrature/. [Google Scholar]
 B.B. Baker, E.T. Copson: The mathematical theory of Huygens’ principle, 3rd edn. Chelsea Publishing, American Mathematical Society 2001, 1987 (1st edition 1939). [Google Scholar]
 G. Green: An essay on the application of mathematical analysis to the theories of electricity and magnetism, facsimiledruck in 100 exemplaren, Berlin, 1889 edn., Nottingham, 1828. [Online]. Available: http://books.google.at/books. [Google Scholar]
 I. Newton: The mathematical principles of natural philosophy. B. Motte, 1729, tranlation by Andrew Motte. [Online]. Available: https://en.wikisource.org/wiki/. [Google Scholar]
 D. Hilbert, R. Courant: Methoden der mathematischen Physik II, Springer, Berlin, 1937. [Google Scholar]
 NIST Digital Library of Mathematical Functions. Release 1.1.9 of 20230315, f. W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, H.S. Cohl, M.A. McClain, eds. [Online]. Available: https://dlmf.nist.gov/. [Google Scholar]
 E. Skudrzyk: The foundations of acoustics, Springer Wien, New York, 1971. [CrossRef] [Google Scholar]
 A. Sommerfeld: Partial differential equations in physics, Academic Press, New York, 1949. [Google Scholar]
 F. Zotter, S. Riedel, L. Gölles, M. Frank: Source code for diffuse sound field synthesis, 2023. [Online]. Available: https://git.iem.at/enimso/2023diffusesoundfieldsynthesisactajupytercode. [Google Scholar]
Cite this article as: Zotter F. Riedel S. Gölles L. & Frank M. 2024. Diffuse sound field synthesis: Ideal source layers. Acta Acustica, 8, 34.
All Tables
Correlation function C(Δx) of the sound pressure at two points spaced by Δx in the ideally isotropic diffuse sound field of D = 1, 2, 3 space dimensions.
Green’s function of the potential equation for D = 1, 2, 3 space dimensions and their derivative.
All Figures
Figure 1 Horizontal map of potential sound energy density (contours), normalized active intensity I/(2 c w) (arrows), and diffuseness w with 200+200 point sources (a) or vertical line sources (b) equally spaced along parallel lines of a 6:1 length:distance ratio, 100 point sources (c) or vertical line sources (d) spaced by 3.6° in azimuth for a circle (salmon), and with a sphere of 480 uncorrelated point sources (e) at Chebyshevtype nodes from [87]; all driven by statistically independent signals of uniform variance. 

In the text 
Figure 2 Layout of uncorrelated source shell S. 

In the text 
Figure 3 Rotationinvariant and shiftinvariant symmetries enforce intensity aligned with surface normal n. 

In the text 
Figure 4 In Newton’s spherical shell theorem, intersection angles only match ϕ_{+} = ϕ_{−} under rotation (middle) or shift (bottom) invariance, not in general (top). 

In the text 
Figure 5 Sectoral intensity to assess isotropy for the 2D circle/3D sphere case (solid) and the 2D parallel line and 3D parallel plane cases (dashed) across observation angle enclosed with the axis of a radial shift of 0% (green), 71% (orange), 87% (blue) of the observer. 

In the text 
Figure 6 Outwards increase of the sound pressure level for the ideal layers (1D: opposing sources, 2D: parallel lines, circle, 3D: opposing planes, cylinder, sphere). 

In the text 
Figure 7 Numerically integrated correlation C(f) for the ideal 2D layers of (a) parallel lines and (b) a circle, the ideal 3D layers of (c) parallel planes, (d) a cylinder, (e) a sphere, in the center x = 0 and offcenter by x = 0.87 with R = 1, r^{2} = 1, and a ∆x = 0.1 of radial (“r”, dotted), axial (“a”, dashdot), or tangential (“t”, dashed) orientation; light to dark solid lines display the ideal isotropic correlations for the dimensions D = 1, 2, 3, cf. Table 1. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.